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hep-th/0101119ITEP-TH-71/00CALT 68-2306CITUSC/00-062HU-EP-00/54SLAC-PUB-8749

Domain Walls, Black Holes, and

Supersymmetric Quantum Mechanics

Klaus Behrndta1 , Sergei Gukovb2 and Marina Shmakovac3

a Institut fur Physik, Humboldt Universitat, 10115 Berlin, Germany

b Department of Physics, Caltech, Pasadena, CA 91125, USA

CIT-USC Center for Theoretical Physics, UCS, Los Angeles, CA 90089, USA

c CIPA, 366 Cambridge Avenue Palo Alto, CA 94306, USA

Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA

Abstract

Supersymmetric solutions, such as BPS domain walls or black holes, in four- and five-

dimensional supergravity theories with eight supercharges can be described by effective

quantum mechanics with a potential term. We show how properties of the latter theory

can help us to learn about the physics of supersymmetric vacua and BPS solutions in these

supergravity theories. The general approach is illustrated in a number of specific examples

where scalar fields of matter multiplets take values in symmetric coset spaces.

January 2001

1 Email: behrndt@physik.hu-berlin.de2 Email: gukov@theory.caltech.edu3 Email: shmakova@slac.stanford.edu

*Work supported by DOE Contract DE-AC03-76SF00515.

1. Introduction and Summary

Supersymmetric vacuum configurations in gauged supergravity theories in four and

five dimensions recently receive a lot of attention due to their relevance to AdS/CFT

correspondence [1], and, more generally, to holography in non-superconformal field theories

and Domain Wall/QFT correspondence [2]. Of particular interest are BPS solutions which

preserve at least four supersymmetries.

In many of such supergravity solutions, the values of scalars and other fields depend

only on a single spatial coordinate. For example, in the case of a BPS black hole this is a

radial coordinate, while for a domain wall this is a transverse spatial coordinate. In any

case, one can integrate out the dynamics in the other directions which are isometries of

the solution and obtain a 0+1 dimensional theory – supersymmetric quantum mechan-

ics – where the distinguished spatial coordinate plays the role of time. This relation to

supersymmetric quantum mechanics was used in certain supergravity solutions, see e.g.

[3]. Interpretation of the attractor flow as supersymmetric quantum mechanics was also

suggested in [4].

In this work we treat both systems in a unified framework of the effective supersym-

metric quantum mechanics with a certain superpotential, so that critical points of the

superpotential correspond to supersymmetric (anti de Sitter) vacua in the supergravity

theory. In the case of domain walls the superpotential is inherited from the original su-

pergravity theory, and in the case of black holes it has the meaning of the central charge

of a black hole. Notice, that in both cases physics requires us to extremize these quanti-

ties. Although the presence of the superpotential is crucial, a lot of interesting questions

can be answered without precise knowledge of its form, simply assuming that it is generic

enough4.

For example, domain walls (and similarly black holes) interpolating between different

minima have interpretation of the RG-flow in the holographic dual field theory on the

boundary. Therefore, physically interesting questions about supersymmetric vacua and

RG-flow trajectories translate into classification of ground states and gradient flows in

the effective quantum mechanics. Following the seminal work of Witten [5], we use the

relation between supersymmetric quantum mechanics and Morse theory to classify these

supersymmetric vacuum configurations. Namely, according to Morse theory, the complex of

critical points with maps induced by gradient flow trajectories is equivalent to the Hodge-de

4 Later we will explain the precise meaning of these words.

1

Rham complex of the scalar field manifold. Notice, that the result is completely determined

by the topology of the scalar field manifold, but not the form of the superpotential.

With these goals and motivation, we start in the next section with the derivation

of supersymmetric quantum mechanics from five-dimensional BPS domain walls which

interpolate between different (AdS) vacua. In the dual four-dimensional theory these

solutions can be interpreted as RG trajectories in the space of coupling constants. We

show that these trajectories are gradient flows with a potential, so-called height function,

given by (the logarithm of) the superpotential in supergravity theory. We also explain

the relation between critical points of the height function and supersymmetric vacua of

the five-dimensional supergravity. In section 3 we extend these results to four-dimensional

gauged supergravity, and also include interaction with hypermultiplets which has not been

studied until recently. In section 4 we draw a parallel with the black hole physics and,

in particular, show how quantum mechanics of the same type appears from the radial

evolution. In this case, the height function is given by the central charge of the black hole.

We also derive effective potentials associated with membranes wrapped over holomorphic

curves in Calabi-Yau compactifications of M theory. After all these systems are reduced

to supersymmetric quantum mechanics, one might hope to achieve classification of the

critical points of the height function by means of Morse theory, cf. [5]. We review the

relevant topology and explain its physical interpretation in section 5. Finally, we put all

the ideas together in section 6 and demonstrate them in a family of simple examples based

on SL(3) symmetric coset spaces.

2. Quantum Mechanics of Domain Walls

BPS domain walls are kink solutions where the scalar fields interpolate between differ-

ent extrema of the supergravity potential and due to the AdS/CFT correspondence, these

solutions are expected to encode the RG flow of the dual field theory. Let us focus here on

the 5-d case with real scalars and postpone the modifications for complex or quaternionic

scalars for the next section. A Poincare-invariant ansatz for the metric reads as follows:

ds2 = e2U(− dt2 + d~x2

)+ dy2 . (2.1)

The function U = U(y) is fixed by the equations of motion coming from the variation of

the action:

S =

∫

M

(R2− 1

2gij∂φ

i∂φj − V)−∫

∂M

K . (2.2)

2

We included a surface term K as the outer curvature which cancels the surface contribution

from the variation of the Ricci scalar. There are no surface terms including the scalars

because they asymptotically extremize the superpotential and hence are constant at the

boundary. The form of the potential

V = 6( 3

4gij∂iW∂jW −W 2

)(2.3)

as a function of the superpotential W is universal for a given dimension and follows from

very general stability arguments [6]. In fact, for complex and quaternionic scalar field

manifolds it is also possible to define a real superpotential W . In this case the supergravity

potential V will have the similar form [7]; see the next section.

The Poincare invariance of the ansatz (2.1) implies that all worldvolume directions

are Abelian isometries, so that we can integrate them out. For our ansatz the Ricci scalar

takes the form R = −20(U)2 − 8U and after a Wick rotation to an Euclidean time we find

the resulting 1-dimensional action5:

S ∼∫

dy e4U[− 6 U2 +

1

2gij φ

iφj + V]. (2.4)

In deriving this expression, the surface term in (2.2) was canceled by the total derivative

term. The equations of motion of this action describe trajectories φi = φi(y) of particles

in the target spaceM with the metric gij . As a consequence of the 5-d Einstein equations,

these trajectories are subject to the constraint

−6 U2 +1

2|φi|2 − V = 0 (2.5)

with |φi|2 = gij∂yφi∂yφ

j. In order to derive the Bogomol’nyi bound we can insert the

potential into (2.4) and write the action as

S ∼∫

dy e4U[− 6 (U ∓W )2 +

1

2

∣∣φi ± 3 ∂iW∣∣2]∓ 3

∫dy

d

dy

[e4U W

](2.6)

leading to the BPS equations for the function U = U(y) and φi = φi(y):

U = ±W , φi = ∓3 gij∂W

∂φj. (2.7)

If these equations are satisfied, the bulk part of the action vanishes and only the surface

term contributes. In the asymptotically AdS5 vacuum this surface term diverges near

5 In our notation, dotted quantities always refer to y-derivatives.

3

the AdS boundary (U ∼ y → ∞) and after subtracting the divergent vacuum energy

one obtains the expected result that the energy (tension) of the wall is proportional to

∆W0 = W+∞ −W−∞.

Our metric ansatz (2.1) was motivated by Poincare invariance which is not spoiled by

a reparameterization of the radial coordinate. We have set gyy = 1, which is one possibility

to fix this residual symmetry. On the other hand, we can also use this symmetry to solve

the first BPS equation W dy = ±dU , i.e. take U as the new radial coordinate. In this

coordinate system the metric reads

ds2 = e2U(− dt2 + d~x2

)+dU2

W 2. (2.8)

Repeating the same steps as before we obtain the Bogomol’nyi equations for the scalars

−φi = gij∂j log |W |3 = gij∂jh (2.9)

which follow from the one-dimensional action

S ∼∫dy[|φi|2 + gij∂ih∂jh

]=

∫dy |φi + gij∂jh|2 + (surface term) (2.10)

where h = 3 log |W |. As before, the field equations are subject to the constraint 12 |φi|2 −

gij∂ih∂jh = 0 and the surface term yields the central charge. Supersymmetric vacua

are given by the extrema of h and the number and type of such vacua can possibly be

determined by using Morse theory where h is called the height function, see [5] and below.

If there are more than two smoothly connected extrema of h, we can build kink

solutions corresponding to domain walls in the 4- or 5-dimensional supergravity. Let us

summarize the different types of domain walls and discuss their implications for the RG

flow and Randall-Sundrum scenario, see also [8]. As long as W 6= 0 at the extremum, we

obtain an AdS vacuum and since the extrema of W are universal (independent of the radial

parameterization), we have to reach the AdS vacuum either near the boundary (U → +∞)

or near the Killing horizon (U → −∞). Obviously, the case U → +∞ corresponds to a large

supergravity length scale and therefore, due to the AdS/CFT correspondence, describes

the UV region of the dual field theory. The opposite happens for U → −∞, which is

related to small supergravity length scales and thus encodes the IR behavior of the dual

field theory. Moreover, extrema of W are fixed points of the scalar flow equations and

translate into fixed points of the RG flow, which can be either UV or IR attractive. The

4

universality of the fixed points of the RG flow, e.g. the scheme independence of scaling

dimensions, translates in supergravity to the fact that the properties of the extrema of the

superpotential are independent of the chosen parameterization of the scalar manifold.

In order to identify the different fixed points we do not need to solve the equations

explicitly; as we will see, they are determined by the eigenvalues of the Hessian of the

height function h. This data depends only on the local behavior of the superpotential near

the fixed point. Let us go back to the BPS equations (2.7) and expand these equations

around a given fixed point with ∂iW∣∣0

= 0 at φi = φi0. The superpotential becomes

W = W0 +1

2(∂i∂iW )0 δφ

iδφj ± . . .

with δφi = φi − φi0, and the cosmological constant (inverse AdS radius) is given by Λ =

−W 20 = −1/R2

AdS. Hence, the scalar flow equations can be approximated by6:

δφi = −(gij∂j∂kW )0 δφk . (2.11)

Next, we can diagonalize the constant matrix (gij∂j∂kW )0 and find

Ωik = (gij∂j∂kW )0 = W01

3∆(i) δik (2.12)

where we absorbed the inverse length dimension into W0. The dimensionless eigenvalues

∆(i) coincide with the eigenvalues of ∂i∂jh. According to the AdS/CFT correspondence

[1], these eigenvalues are the scaling dimensions of the corresponding perturbations in the

dual field theory. Namely, in a linearized version, the equations of motion for the scalars

become ∂2φi−M ijφj = 0 and the mass matrix readsM i

j = ∂i∂jV∣∣0

= W 20 ∆(i)(∆(i)−4)δij

or, measured in the units of W0, the mass formula becomes

(m(i))2 = ∆(i)(∆(i) − 4) . (2.13)

Consequently, near the AdS vacuum we find a solution to (2.11)

U = (y − y0)W0 , δφi = e−13 ∆(i)W0(y−y0) = e−

13 ∆(i)U . (2.14)

This approximate solution is, of course, valid only if δφi = φi−φi0 → 0 in the AdS vacuum

where U → ±∞ and therefore all eigenvalues ∆(i) have the same sign: ∆(i) > 0 for UV

6 For definiteness we took the upper sign convention.

5

fixed points (U → +∞), or ∆(i) < 0 for IR fixed points (U → −∞). Equivalently, UV

fixed points are minima of the height function h whereas IR fixed points are maxima. For

this conclusion we assumed that the scalar metric has Euclidean signature and W0 > 0. It

is important to notice that in the definition of the scaling dimensions the matrix Ωij has

one upper index and one lower index. It is straightforward to consider also the possibilities

W0 < 0 and/or timelike components of the scalar field metric. Note, the sign ambiguity in

the BPS equations (2.7) interchanges both sides of the wall, i.e. it is related to the parity

transformation y ↔ −y, which also flips the fermionic projector onto the opposite chirality.

The eigenvalues of Ωik of different signs mean that the extremum is IR-attractive

for some scalars and UV-attractive for the other and, therefore, is not stable (a saddle

point of h). Moreover, using these saddle points to connect two maxima/minima of h

would violate the proposed c-theorem for domain walls [9,10,11,12]. Namely, multiplying

eq. (2.9) by gikφk one obtains

−h = −φi∂ih = gij φiφj ≥ 0 . (2.15)

Therefore, along the flow, the height function h has to behave strictly monotonic and at

the extrema it corresponds to the central charge of the dual conformal field theory [13]:

cCFT ∼ R3ADS = 1/|W0|3 = e−h0 . Recall that in our sign convention larger values of the

radial parameter U correspond to the UV region and are minima of the height function h.

If we start with the UV point (U = +∞) and go towards lower values of U , the c-theorem

states that h has to increase, either towards an IR fixed point (maximum) or towards

a positive pole in h (W 2 → ∞), which is singular in supergravity and corresponds to

cCFT = 0. On the other hand, if we start from an IR fixed point (U = −∞) and go towards

larger values of U , due to the c-theorem h has to decrease, either towards a minimum (UV

fixed point) or towards a negative pole (W 2 → 0), which is not singular in supergravity

and corresponds to flat spacetime, cCFT → ∞. An example is the asymptotically flat

3-brane, where the height function parameterizes the radius of the sphere, which diverges

asymptotically (indicating decompactification) and runs towards a finite value near the

horizon which is IR attractive in our language.

In summary, there are the following distinct types of supergravity flows, which are

classified by the type of the extremum of the height function or superpotential. Depending

on the eigenvalues of the Hessian of h, the extrema can be IR attractive (negative eigenval-

ues), UV attractive (positive eigenvalues) or flat space (singular eigenvalues). Generalizing

6

the above discussion and allowing also possible sign changes in W , the following kink so-

lutions are possible (in analogy to the situation in four dimensions [8]):

(i) flat ↔ IR

(ii) IR ↔ IR

(iii) IR ↔ UV

(iv) UV ↔ UV (singular wall)

(v) UV ↔ singularity (W 2 =∞).

Note, there is no kink solution between a UV fixed point and flat space, because the c-

theorem requires a monotonic h-function and the UV point corresponds to a minimum of h,

whereas the flat space case is a negative pole. Moreover, if there are two fixed points of the

same type on each side of the wall, W necessarily has to change sign implying that the wall

is either singular (pole in W ) or one has to pass a zero of W . In addition, between equal

fixed points no flow is possible (that would violate the c-theorem) and therefore this case

describes a static configuration, where the scalars do not flow. This is also what we would

expect in field theory, where the RG-flows go always between different fixed points. Recall,

although a zero of W means a singularity in h, the domain wall solution can nevertheless

be smooth. In models with mass deformations (∆(i) = 2), type (v) walls appear generically

for models which can be embedded into maximal supersymmetric models, whereas models

allowing type (iv) walls typically can not be embedded into maximal supersymmetric

models7, an explicit example is discussed in [14].

In the Randall-Sundrum scenario one is interested in the case (ii), because in this case

the warp factor of the metric vanishes exponentially on each side of the wall.

3. Potentials from Gauged Supergravity

Extrema of the supergravity potential V are vacua of the theory, but not all extrema

correspond to stable vacua. Instead, one can show [6] that stable vacua are extrema of a

superpotential W which defines the supergravity potential in d dimensions to be

V =(d− 2)(d − 1)

2

( d− 2

d− 1gij∂iW∂jW −W 2

).

Once again, the corresponding scalar flow equations look like φi = −gij∂jh with the height

function

h = (d− 2) log |W |.7 Maximal supersymmetric models typically have only one UV extremum.

7

We will now derive the real superpotential W for different models. Depending on the num-

ber of unbroken supercharges, only special superpotentials can appear in supersymmetric

models. Our primary interest are supergravity duals of field theories with 4 unbroken

supercharges. Therefore the scalars on the supergravity side are part of vector, tensor or

hyper multiplets that parameterize the product space:

M =MV/T ×MH.

Known potentials are related to: (i) gauging isometries of M or (ii) gauging the global R-

symmetry. In the first case, the scalars and fermions become charged whereas in the second

case the scalars remain neutral. As a consequence of the gauging, the supersymmetry

variations are altered and flat space is, in general, not a consistent vacuum. If the potential

has an extremum, it is replaced by an AdS vacuum, which was not welcomed in the early

days of supergravity, but fits very well in the AdS/CFT correspondence and the holographic

RG flow picture. Let us discuss the different cases in more detail.

3.1. Gauged Supergravity in 5 Dimensions

Supergravity in five dimensions needs at least 8 supercharges, and scalar fields can

be part of vector-, tensor- or hypermultiplets. Each (abelian) vector multiplet contains a

U(1) vector Aiµ, a gaugino λi and a real scalar φi (i = 1, . . . , nV ). On the other hand, a

hypermultiplet includes two hyperinos ζu and four real scalars qu (u = 1, . . . , 4nH). Finally,

the gravity multiplet has besides the graviton, the gravitino ψAm and the graviphoton A0µ.

On the vector multiplet side, supersymmetry is a powerful tool in determining allowed

corrections. In fact, all couplings entering the Langrangian are fixed in terms of the cubic

form [15]

F =1

6CIJKX

IXJXK . (3.1)

In a Calabi-Yau compactification of M-theory, the fields XI (I = 0, . . . , nV ) are related

to the Kahler class moduli M I by a rescalling XI = MI

V1/3 with the Calabi-Yau volume

V = 16CIJKM

IMJMK and the constants CIJK are the topological intersection numbers

[16]. The scalar fields φa(XI ) parameterize the spaceMV defined by F = 1 and the gauge-

and scalar-couplings are given by

GIJ = −1

2(∂I∂JF )F=1, gij = (∂iX

I∂jXJGIJ )F=1. (3.2)

8

Much less is known on the hypermultiplet side. The four real scalars of each hyper-

multiplet are combined to a quaternion and parameterize a quaternionic mannifold MH.

From the geometrical point of view, the hypermultiplet sector in four and five dimensional

supergravity is the same, see [17] for further details. In any compactification from string

or M-theory there is at least one hypermultiplet – the so-called universal hypermultiplet

that contains Calabi-Yau volume V. This name is a little bit misleading since when nH > 1

there is no unique way to single out one direction on a general quaternionic manifold [18].

As long as we have only this single hypermultiplet, its geometry is given by the coset spaceSU(2,1)U(2) . But after including further hypermultiplets, quantum corrections will deform the

space in a way which is rarely known.

Now let us turn to potentials resulting from gauged isometries. Both manifolds,MV

and MH , have a number of isometries [19], which can be gauged [20,21]. However the

flow equations of the scalars are not sensible to gauged isometries ofMV ; it yields only an

additional “D-flatness” constraint8 [22,21]. More interesting is the gauging of isometries

ofMH ; see [23,7,24] for explicit examples. This is a quaternionic space, which implies the

existence of three complex structures and an associated triplet of Kahler forms Kx. The

holonomy group is SU(2)×Sp(nH) and the Kahler forms have to be covariantly constant

with respect to the SU(2) connection. The isometries are generated by a set of Killing

vectors kuI

qu → qu + kuI εI

and the required gauge covariant derivatives become dqu → dqu + kuIAI . In order to keep

supersymmetry, this gauging has to preserve the quaternionic structure, which means that

the Killing vector has to be tri-holomorphic (in analogy with the holomorphicity in N=1

supergravity). This is the case if we can express them in terms of a triplet of Killing

prepotentials P xI (with the SU(2) index x = 1, 2, 3)

Kxuvk

vI = −∇uP xI ≡ −∂uP xI − εxyzωyuP zI . (3.3)

Here ωyu are the SU(2) connections related to the Kahler forms by Kxuv = −∇[uω

xv]. They

can be also expressed in terms of the complex structures and the quaternionic metric as:

Kxuv = (Jx) ru hrv, and using

∑x(Jx) ru (Jx) vr = −3 δ vu we can write the Killing vectors as

kuI = −∑x huv(Jx) rv ∇rP xI . Next, introducing an SU(2)-valued superpotential

W BA ≡W x (iσx) BA with : W x = XIP xI = XI(φ)P xI (q) (3.4)

8 See also below for the similar situation in 4 dimensions.

9

(σx are the Pauli matrixes) the fermionic supersymmetry variations [21] become

δψAm =DmεA − i

3W AB Γmε

B ,

δλAi =− i

2

[Γm∂mφ

i εA − 2i gij∂jWAB εB

],

δζα =− i√2V Aαu

[Γm∂mq

u − 2huv(Jx) rv ∇rW x)]εA

(3.5)

where A,B are SU(2) indices and ∂i ≡ ∂∂φi

. We dropped the gauge field contributions,

since they are not important for flat domain wall solutions; they would be important e.g.

for a worldvolume geometry R × S3. For supersymmetric vacua, W BA has to become

extremal (∂iWx = ∇uW x = 0, for all x = 1, 2, 3). One can show [7], that the SU(2)

phase of W does not contribute to the scalar flow equations and is absorbed by the SU(2)

connection entering the covariant derivative Dm. Moreover, combining all scalars (φi, qu)

that parameterize the space M = MV ×MH with the metric gij = diag(gij , huv) one

recovers the BPS equation (2.7) with the real-valued superpotential

W 2 =∑

x

W xW x . (3.6)

There is one especially simple example, where the superpotential is only U(1) valued,

i.e. the Killing prepotential has only one, say, P 3I component. In this case the SU(2)

covariant derivative in (3.3) becomes a partial derivative and we have the freedom to shift

the Killing prepotential by any constant P 3I → P 3

I + αI . In fact, one can even set the

Killing prepotential to zero and keep only the constants αI , which are the analogs of the

FI-terms in field theory. As a consequence, the Killing vectors vanish as well as the charges

of the scalars. But still, we have a non-trivial potential giving a mass to all vector scalars.

In this case the superpotential becomes

W (3)(φi) = (αIXI)F=1 , (3.7)

which is manifestly real valued. Due to the constraint F = 1 this potential yields an AdS

vacuum, where generically all vector-scalars are fixed and the moduli space MV of vacua

is lifted except for a discrete set of extremal points of W (3). From the 5-d supergravity

perspective this model has been discussed in [25], and important for the RG-flow is the

property [15,26]

∂i∂jW(3) =

2

3gijW

(3) + Tijk∂kW (3) . (3.8)

10

This relation implies that all scaling dimensions, as defined in (2.12), are ∆(i) = +2 and

fulfill a sum rule [27]:∑

i ∆(i) = 2n, with n = dimMV . Therefore, the flow is generated by

mass deformations in the field theory and the positive sign indicates that all fixed points

are UV attractive; IR attractive critical points are excluded for this model [22,28]. This

model can be obtained from Calabi-Yau compactification of M-theory in the presence of

non-trivial G-fluxes parameterized by αI [23] and the superpotential can be written as

[29,30]

W (3) =

∫

CY

K ∧Gflux (3.9)

where K is the Kahler 2-form. However this compactification yields an un-stabilized

Calabi-Yau volume and, as long as we treat it as a dynamical field [31,30], this compacti-

fication does not give flat space or AdS vacua. Nevertheless this run-away problem can be

avoided by more general hypermultiplet gauges, see [7]. It would be interesting to derive

the general SU(2)-valued superpotential in the same way from M-theory. For a recent

study of W x (with W 3 = 0 but W 1 6= 0 and W 2 6= 0) see [32].

3.2. Gauged Supergravity in 4 Dimensions

Supergravity in 4 dimensions needs at least 4 supercharges and allows for more general

(holomorphic) superpotentials which are not related to gauged isometries. In the generic

case, these models have an AdS vacuum with a dual 3-d field theory with only 2 (unbroken)

supercharges; for domain wall solutions see [8].

However, if we again focus our attention on models with 8 supercharges, potentials

have to be related to gauged isometries. The main difference from the situation in 5 dimen-

sions concerns the vector multiplet side. Since vector fields in four dimensions have only

two on-shell degrees of freedom, each vector multiplet has to contain two scalars (in order

to complete the bosonic degrees of freedom). These two real scalars can be combined into

a complex scalar zi and supersymmetry requires that they parameterize a special Kahler

space, see [17] for a review. On the other hand, the scalars qu entering the hypermultiplets

parameterize again a quaternionic space and the gauging of the corresponding isometries

goes completely analogous to the case in 5 dimensions. This time, however, it is reason-

able to use the symplectic notation of special geometry and we will use the holomorphic

symplectic section V =(XI(z), FI (z)

)with the symplectic product defining the Kahler

potential:

e−K/2 = 〈V,V〉 = i(XIFI −XIF I). (3.10)

11

Then, we obtain a similar superpotential, cf. eq. (3.4):

W BA ≡W x (iσx) BA with W x = XI P xI = XI(z)P xI (q) . (3.11)

Notice, now XI = XI(z) is a complex field. In N=2 supergravity in 4 dimensions one

also introduces a symplectic vector Fµν = (F Iµν , GI µν) for the gauge fields. Both gauge

fields are related to each other and we took the freedom to transform all gauge fields to

F Iµν and therefore only the XI component of the section enters W . As in 5 dimensions we

can express the supersymmetry variations [33,17] in terms of the superpotential (3.11)

δψAm =DmεA − i

3eK/2 W A

B γmεB ,

δλAi =− i

2

[γm∂mz

i εA + 2i eK/2 gij∇jWA

B εB + kiIXIeK/2εA

],

δζα =− i√2V Aαu

[γm∂mq

u − 2 eK/2 huv(Jx) rv ∇rW x)]εA

(3.12)

with ∇jW x = P xI (q)(

∂∂zj + ∂K

∂zj

)XI(z) as a Kahler covariant derivative and ∇rW x denotes

the SU(2) covariant derivative, see (3.3). We included also a possible gauging of MV

related to the Killing vector kiI and remarkably this gauging affects only the gaugino

variation δλAi, but not the gravitino variation δψAm.

As usual the fermionic projector is derived from the timelike gravitino variation δψA0

and using the metric ansatz (2.8) this projector becomes

γU εA ± i W

AB

|W | εB = 0 (3.13)

and we define a real superpotential by

W 2 = eK |W |2 =∑

x

eKW xWx

=∑

x

eKXIXJP xI P

xJ (3.14)

(the hyper scalars qu are 4nH real fields so that the Killing prepotentials are real). Note,

the projector (3.13) contains both, the SU(2) phase as well as the U(1) phase related to

the complex fields XI and in order to ensure the vanishing of the radial gravitino variation

δψAU , both phases have to be absorbed into the SU(2) (resp. U(1)) Kahler connection.

This can impose further constraints.

12

Using the projector it is straightforward to show that the gaugino variation yields two

equations9: a D-flatness constraint kiIXI

= 0 and, after fixing the sign ambiguity in the

projector, the expected flow equation for the scalars zi:

zi = −gij∂zj logW 2 . (3.15)

Here we use: ∂iW2 = 2W∂iW =

∑x e

KW x(∂i + ∂iK)Wx

=∑

xW2

|W |2Wx∇iW

x. Finally,

following the steps done in 5 dimensions, see also [7], the hyperino variation yields the

same flow equation

qu = −huv∂v logW 2 .

As before, we can again consider the special case, where the Killing prepotentials have

only one component (e.g. P 1I = P 2

I = 0, P 3I 6= 0) and can be shifted by arbitrary constants

αI . The analog of (3.7) is now the complex superpotential [17]

W (3) = αIXI(z) .

Recall that this form is related to the special symplectic basis, where all GI µν gauge fields

have been dualized to F Iµν gauge fields. In general, the superpotential has the covariant

form [34,31]

W (3) = αIXI − βIFI . (3.16)

It allows for AdS vacua which are in one-to-one correspondence with solutions to the

attractor equations, which determine extrema of the supersymmetry central charge [35,36].

One can show that all these extrema are UV attractive [28] (due to a similar relation as

(3.8)) and, therefore, cannot give regular domain wall solutions. Like in 5 dimensions,

this superpotential is related to a compactification in the presence of fluxes. This time,

however, it is type IIB Calabi-Yau compactification and the superpotential can be written

as [31,29]

W (3) =

∫

CY

Ω ∧Hflux

where Ω is the holomorphic (3,0)-form and Hflux = HRRflux + τHNS

flux with τ as the com-

plexified scalar field of type IIB string theory. Supersymmetric vacua corresponding to the

9 In this matrix equation, the coefficient in front of each Pauli matrix has to vanish. In addition,

in all these calculations one has to keep in mind that the γ-matrix implicitly contains a W factor

due to its curved index.

13

minima of this superpotential have been recently studied in [32]. Moreover, the authors

of [32] pointed out a relation between these supersymmetric vacua and attractor points

of N = 2 supersymmetric black holes. Namely, they demonstrated the equivalence of the

supersymmetry conditions ∇iW x = 0 to the attractor equations that determine the values

of the scalar fields zi at the horizon of a black hole. We elaborate this relation in the next

section from the standpoint of the effective quantum mechanics, whereas in the rest of this

section we discuss ‘attractor’ interpretation of the other supersymmetry condition:

∇uW x = XI(∂uP

xI + εxyzωyuP

zI

)= 0 . (3.17)

The idea is that the supersymmetry variations of the hyperino (3.5) (or (3.12)) de-

scribe supersymmetric vacua of N = 2 gauged supergravity as well as BPS instantons (in

ungauged supergravity) responsible for non-perturbative corrections to the metric on the

moduli space of hypermultipletsMH . In order to see the relation more precisely, consider

supergravity theory obtained from Calabi-Yau compactification of M theory (or type IIA

string theory). Then, the instantons in questions are constructed from (bound states of

membranes and) Euclidean 5-branes wrapped on the Calabi-Yau manifold [37]. Put dif-

ferently, this BPS configuration can be understood as a five-brane wrapped around the

Calabi-Yau space with world-volume 3-form tensor field turned on.

Apart from the number of 5-branes, the instanton is characterized by the membrane

charge αu that takes values in the homology lattice H3(CY,ZZ). For a given set of the

charges, one can construct a spherically symmetric solution qu(r) which preserves half the

supersymmetry (3.5) and behaves like10 [37,38]:

qu(r) ∼ αu3r3

+ const, r →∞ . (3.18)

When r varies from r =∞ to r = 0, the scalar fields qu go to the fixed values determined

by the charges αu, similar to the attractor mechanism [35]. It was shown in [37], that the

radial evolution of the scalar fields qu(r) is described by null-geodesics in the effective 0+1

dimensional theory with target space MH. Up to boundary terms, we expect the action

of this effective theory is equivalent to the action of supersymmetric quantum mechanics

(2.10) with the superpotential (3.4) (or (3.11)). Thus, we claim that the flow on the moduli

space of the hypermultiplets is governed by W x.

10 For the sake of concretness, we consider five-dimensional case obtained from compactification

of M theory. Reduction to four dimensions is straightforward.

14

Since the instanton solution is expected to be smooth at r = 0, from the equation

(3.5) (with the U(1)R gauge field contribution included) we find that the fixed points of

the flow on the moduli space of hypermultiplets are characterized by the condition (3.17):

XIkuI = 0 . (3.19)

This simple condition means that the Killing vectors obey certain linear relations at the

fixed points in the space MH . Note, that in the hypermultiplet version of the attractor

mechanism we have fixed limit cycles rather than fixed points11. It would be interesting

to better understand the physics and the geometry of these fixed cycles in MH [39].

Intuitively, the hypermultiplet attractor equation (3.19) could be expected by analogy

with N = 2 BPS black hole solutions that exhibit enhancement of supersymmetry at the

horizon. For a similar reason, one might expect enhancement of supersymmetry at the

center, r = 0, of the BPS instanton constructed from a bound state of a membrane

wrapped around a special Lagrangian cycle and a five-brane wrapped on the entire Calabi-

Yau space. As we explained in the earlier sections, a supersymmetric vacuum is given by

the extremum of the superpotential (3.17) which, in turn, is equivalent to (3.19).

4. Black Hole – Domain Wall Correspondence

Similarity between physics of black holes of ungauged supergravity and the domain

wall solutions in gauged supergravity with gauging of vector multiplets [40,41] is based

on the fact that both theories can be described by the same one-dimensional effective

Lagrangian. In both cases the solution is fixed by the same set of attractor equations

[35,36] and the superpotentialW of gauged supergravity corresponds to the supersymmetry

central charge Z.

The D=4 N=2 action of ungauged supergravity in terms of special geometry has a

well known form:

L =R

2−Gij∂µzi∂µzj − huv∂µqu∂µqv − ImNΛΣFΛFΣ −ReNΛΣFΛ∗FΣ (4.1)

where z as in Section (3.2) are the complex scalars of vector multiplets parameterizing

special Kahler manifold MV with metric Gij = ∂2K

∂zi∂zjand Kahler potential K. Their

11 We would like to thank A. Strominger for pointing this out to us and drawing our attention

to the hypermultiplet version of the attractor mechanism as we describe it here.

15

kinetic term can be written in the form similar to the one in (2.2): gijdφidφj = Gij∂z

i∂zj .

The vector couplings ImN and ReN depend only on scalar fields z. Real scalars qu belong

to hypermultiplets and parameterize the quaternionic manifoldMH with the metric huv.

In ungauged supergravity the hypermultiplets are decoupled from the theory and we will

consider only vector multiplet scalars. We will choose the following ansatz for D=4 extreme

black hole metric [42]:

ds2 = −e2Udt2 + e−2U

(dτ 2

τ 4+

1

τ 2dΩ2

2

).

This assumption leads to one-dimensional effective action of the familiar form:

Seff ∼∫ ((

dU

dτ

)2

+ gijdφi

dτ

dφj

dτ+ e2UV (φ, p, q)

).

The potential V depends on the symplectic covariant electric and magnetic charges (pI , qI )

for symplectic vector Fµν = (F Iµν , GI µν) and can be identified with the symplectic invariant

form I1 in terms of the complex central charge (the graviphoton charge) Z = qIXI − pIFI

with the symplectic section (XI , FI):

V (φi, p, q) = I1 = |Z(z, p, q)|2 + |∇iZ(z, p, q)|2 (4.2)

where ∇i is a Kahler covariant derivative. Action (4.1) can be easily transformed to the

standard form of Bogomol’nyi bound:

L =

(dU

dτ± eU |Z|

)2

+∣∣dzdτ± eUGij∇jZ

∣∣2 ± d

dτ[eUZ]

with the equations of motion:

dU

dτ= ±eU |Z|, dzi

dτ= ±eUGij∇jZ. (4.3)

Using the same construction as in Section 3.2 for (3.15) we finally derived the equation

for the scalar field z:dzi

dU= Gij∂j logZ2.

This form is consistent with one-dimensional supersymmetric quantum mechanics.

16

For the black hole solutions in D=5 ungauged supergravity [36,42,38] the situation is

similar. The action of D=5 N=2 ungauged supergravity coupled to vector multiplets has

a form:

S5 =

∫R

2− 1

4GIJF

IF J − 1

2gij∂φ

i∂φj − e−1

48εµνρσλCIJKF

JµνF

JρσA

Kλ .

Scalar fields φi are defined through the constraint (3.1), F = 16CIJKX

IXJXK = 1, and

the gauge coupling GIJ and gij are given by (3.2) and depend only on XI and CIJK . The

ansatz for the D=5 black hole solutions is:

ds2 = −e−4Udt2 + e2U(dr2 + r2dΩ2

3

), (4.4)

and the ansatz vector fields is: GIJFItr = 1

4∂rKJ where KI = kI + QIr2 and QI are black

hole electric charges.

Using a new radial variable τ = 1r2 and equations of motion it is easy to reduce D=5

action to the effective D=1 dimensional action:

Seff ∼∫ (

3

(dU

dτ

)2

+1

2gij

dφi

dτ

dφj

dτ+ e4U 1

12V (φ, p, q)

)(4.5)

where the potential V comes from the kinetic term for the vector fields

V =3

2QIQJG

IJ = Z2 +3

2gij∂Zi∂Zj (4.6)

where Z = XIQI is a central charge. In (4.6) we used special geometry relations (see for

example [43]): gij = GIJXI,iX

J,j = −3CIJX

I,iX

J,j and gijXI

,iXJ,j = GIJ − 2

3XIXJ where

XI,i = ∂XI

∂φi .

The effective action (4.6) is easily changed to the familiar BPS form:

L = 3

(dU

dτ± 1

6e2UZ

)2

+1

2

∣∣dφi

dτ± 1

2e2U∂iZ

∣∣2 ∓ 1

2

d

dτ[e2UZ]

where |dφidτ |2 = gijdφi

dτdφj

dτ . Once again we have a system of linear differential equations of

the form:∂U

∂τ± 1

6e2UZ = 0, gij

∂φj

∂τ± 1

2e2U ∂Z

∂φi= 0.

Using the first equation to introduce the new radial variable dU = 16e

2Udτ we get the

familiar equation for the gradient flow in a scalar field manifoldMV :

∂φi

∂U= gij∂j logZ3.

17

This consideration shows that both cases – black holes of ungauged supergravity

coupled to vector multiplets and domain walls of gauged supergravity can be treated using

the same approach of one-dimensional supersymmetric quantum mechanics.

The first interesting question connected to the above discussion is the form of the

black hole potential (4.6) . The same potential appears as a result of M-theory compact-

ification on Calabi-Yau threefolds in the presence of non-trivial G-fluxes [23,30,21] and

corresponds to gauging of the universal hypermultiplet of D=5 dimensional theory. An

electrically charged five dimensional black hole corresponds to a membrane in M-theory

wrapped around 2-cycles of Calabi-Yau manifold in the process of compactification. We

can formally regard the M-theory compactification in the presence of the membrane-source

and corresponding flux.

The 11-dimensional supergravity theory is described by the action:

S11 =1

2

∫

M11

(√−gR− 1

2G ∧ ∗G− 1

6C ∧G ∧G

)

where C is a 3-form field with 4-form field strength G(4) = dC. In the presence of an

electrically charged membrane source Bianchi identities and equations of motion forD = 11

theory are:

dG = 0 (4.7)

d?G = d∗G+1

2G ∧G = 2k2

11(∗J) (4.8)

where ?G = ∂L∂G ,

∗ is Hodge-dual and J is a source current with the corresponding Noether

“electric” charge

Q =√

2k11

∫

M8

(∗J) =1√

2k11

∫

S7

(?G)(7)

where M8 is a volume and S7 is a sphere around the membrane. A formal solution to the

equations of motion (4.8) reads:

?G =∗ G+1

2C ∧ G =

√2k11Q

ε7Ω7

where ε7 is a volume form and Ω7 is a 7-volume. This solution corresponds to the vector

potential of the form:

C ∼ Q

r6ω3 (4.9)

where r is a D = 11 transverse distance to the membrane and ω3 is the membrane volume

form. Compactification from D = 11 dimensions down to D = 5 leads to the configuration

when the membrane is wrapped around compact dimensions.

18

A natural splitting of moduli coordinates Ma ⇒ (XI = MI

V1/3 ;V) and the condition

for function (3.1) : F (X) = 1 defines (h1,1 − 1)-dimensional hypersurface on the Calabi-

Yau cone and (h1,1 − 1) independent coordinates φa(XI ) (special coordinates) on this

hypersurface define vector multiplet moduli space with Kahler metric GIJ (3.2) :

GIJ (X) =i

2V

∫ωI ∧ ∗ωJ = −1

2∂I∂J logF (X)|F=1. (4.10)

A solution (4.9) of the equations of motion take the form:

G =1

V1

r3dt ∧ dr ∧ αIωI . (4.11)

Here r is a D = 5 transverse radial coordinate and V is a Clabi-Yau volume, and “electric”

charges αI = GIJαJ are

αI =

∫

C(4)I×S3

(∗G)

where C(4)I, I = 1, ..., h1,1 are 4-cycles in the Calabi-Yau manifold. The nontrivial flux

of this form leads to the appearance of the nonzero scalar potential. From the G∧∗G term

of the action it follows that∫

M11

G ∧∗ G =

∫

M11

√gstr

r6

1

V2αIαJωI ∧ ∗ωJ = 2

∫

M5

√gE

r6

1

V2αIαJG

IJ(X) (4.12)

here we use the relation√gstr = V−5

3√gE . The effective potential for D = 5 is:

V5 =1

r6

1

V2αIαJG

IJ(X).

This D=5 potential V5 explicitly depends on r and fells with the distance to the source

so that in this case there is no run-away Calabi-Yau volume as in [23,30].

In the effective one-dimensional theory (4.5) U plays a role of a dilaton and the

appearence of the effective potential (4.6) can be understood as a result of gauging of an

additional axionic shift symmetry.

The consideration of Section 3 shows that hypermultiplets can also play a very impor-

tant role in the physics of black holes. The difference between a black hole solution and a

domain wall solution is in the presence of non-trivial vector fields in the black hole case.

Those fields should be added to the supersymmetry variations (3.5) (see [21]):

δψAm =DmεA +

i

8XI(Γ

npm − 4δnmΓp)F Inpε

A − i

3W AB Γmε

B ,

δλAi =− i

2

[Γm∂mφ

i εA − 2i gij∂jWAB εB

]+

3

8gij∂jXIΓ

mnF ImnεA ,

δζα =− i√2V Aαu

[Γm∂mq

u − 2huv(Jx) rv ∇rW x)]εA.

(4.13)

19

It is easy to see that in ungauged supergravity hypermultiplets are decoupled from

the theory and do not affect the solution. It may not be the case when some gauging is

present.

An application of Morse theory to the black hole physics can have interesting conse-

quences. It is possible to consider a black hole solution in gauged supergravity with vector

multiplet gauging [44] . In this case hypermultiplets are also decoupled and, unfortunately

(as we will see later, rather, predictably), the BPS solutions contain naked singularities

or blow up near the horizon. This means that in this theory it is impossible to find a

black hole with a regular horizon embedded in the AdS5. On the other hand, application

of Morse theory to this case predicts existence of only one critical point (or several of the

same type) and the absence of a second non-trivial vacuum (see Section 5) is justified.

The case of the black hole solution with hypermultiplet gauging is much more com-

plicated. In this case, hypermultiplets are no longer decoupled from the theory and the

hypermultiplet gauging can lead to the appearance of a second non-trivial critical point.

It will be interesting to consider such solutions in the future work.

5. Morse Theory and Vacuum Degeneracy

In the previous sections we studied the effective dynamics of BPS solutions in N = 2

five-dimensional supergravity which preserve the SO(3, 1) (or SO(4)) subgroup of the

Lorentz symmetry, as well as the effective dynamics of the similar solutions in four dimen-

sions. The examples — such as supersymmetric domain walls or spherically symmetric

static black holes — correspond to solutions where space-time metric is a function of a

single coordinate. Effective dynamics of a BPS state with these properties turns out to

be supersymmetric quantum mechanics of the form (2.10), cf. [45,46,47]. The nature of

the BPS state is perfectly indifferent. For all such BPS states it is true that the solution

represents a gradient flow of the height function h between two critical points in the scalar

field manifoldM. For example, in five-dimensional gauged supergravity coupled to a cer-

tain number of vector multiplets M = MV is a Riemannian manifold parameterized by

scalar fields φA from the vector multiplets and h is (logarithm of) the superpotential W .

Moreover, if this theory is obtained from compactification of M theory on a Calabi-Yau

three-fold, thenMV is just the Kahler structure moduli space of this Calabi-Yau manifold,

and h has a microscopic interpretation in terms of a G-flux [23,30].

20

In any case, the study of BPS objects described above and the classification of su-

persymmetric vacua in D = 4 and D = 5 supergravity boils down to a simpler problem

in supersymmetric quantum mechanics with the effective action (2.10). This connection

to supersymmetric quantum mechanics allows one to address many interesting physical

questions studying more elementary system. For example, Klebanov and Tseytlin used

this relation to study supergravity duals of the RG-flows in SU(N) × SU(N +M) gauge

theories [3]. In this section we will discuss another application of this relation.

For a compact space M, it has been shown by Witten [5] that supersymmetric quan-

tum mechanics with the action (2.10) is just Hodge-de Rham theory of M. Therefore,

assuming it is also the case for certain (non-compact) scalar field manifolds M that oc-

cur in supergravity, we can use topological methods — in particular, Morse theory — to

classify possible BPS states and supersymmetric vacua they connect. As we explained

above, our results are quite generic irregardless of the physical nature of a given BPS

solution; they work equally well for BPS domain walls, for spherically symmetric black

holes, or any other BPS object with the effective action (2.10). We only make a couple of

assumptions necessary in the following. First, we requireM to be a Riemannian manifold,

although sometimes it comes with some additional structure (e.g. complex or quaternionic

structure) corresponding to additional (super-)symmetry in the problem. One has to be

very careful with this assumption in the cases when M has singularities, in particular, in

Calabi-Yau compactifications of M theory. Second, we assume that the space M is either

compact or it has the right behaviour at infinity, so that the topological methods of Morse

theory are reliable12.

Given a Riemannian manifoldM, let h:M→ R be a ”good” Morse function of C∞-

class. This condition means that every point in M is either a regular point where dh 6= 0

or an isolated critical point p ∈ M where dh = 0 and h can be written as:

h(φA) = h(p) −k∑

A=1

(φA)2 +n∑

A=k+1

(φA)2 (5.1)

in some neighborhood of p.

The number k in (5.1) is called the Morse index of a critical point p and is denoted

by µ(p). It turns out that µ(p) has a nice physical interpretation in supergravity. Recall,

12 For an analog of Morse theory in complex non-compact geometry that admits a holomorphic

torus action see [48].

21

that the critical points of h correspond to supersymmetric vacua. Moreover, according to

the analysis of section 2, the eigenvalues of the Hessian of h determine the type of the

attractor point. In particular, critical points with zero Morse index are UV attractive. On

the other hand, critical points with µ = n are IR attractive. In general, a critical point

p is of mixed type, 0 < µ < n. In models with only vector multiplets, the superpotential

depends generically on all scalars and such mixed critical points do not correspond to

stable vacua, see discussion after (2.14).

Therefore, in order to classify possible supersymmetric vacua, one has to know how

many points have a given Morse index. Morse theory provides a nice answer to this problem

in terms of the topology of the scalar field manifold M. Before we state the result let us

introduce a few more notions which will be convenient in the following.

As we explained in the previous sections, a supersymmetric domain wall or a spheri-

cally symmetric black hole corresponds to a gradient flow13 of h from one critical point p

to another critical point q. Consider the “moduli space of these BPS solutions” M(p, q).

Namely, for a pair of critical points p and q we define:

M(p, q) = φ: R→M | dφdt

= −∇h, limt→−∞

φ(t) = p, limt→+∞

φ(t) = q/ ∼

to be the moduli space of the gradient trajectories from p to q modulo the equivalence

relation φ(t) ∼ φ(t+ const). In general, M(p, q) is not a manifold, though perturbing it a

bit we can always assume that it is a manifold (or a collection of points, if we have a finite

number of distinct gradient trajectories).

A classical result in Morse theory asserts that the real dimension of M(p, q) is given

by:

dimM(p, q) = µ(p) − µ(q) − 1. (5.2)

It immediately follows that non-singular BPS domain walls (or spherically symmetric black

holes) of types (ii) and (iv) interpolating between two vacua of the same kind do not exist

in theories where h is a global Morse function, cf. the discussion in the end of section 2.

Indeed, if there are two critical points of the same type (i.e. either both IR or both UV),

then the virtual dimension (5.2) becomes negative. Another physical result that follows

from (5.2) is that domain walls of type (iii) connecting an IR attractive point and a UV

attractive point come in (n− 1)-dimensional families.

13 It is this place where we use the Riemannian metric on M to define a gradient vector field

∇h.

22

It is curious to note that BPS solutions interpolating between mixed (IR/UV) vacua

also appear in Morse theory as boundary components ofM(p, q). For example, codimension

1 boundary of M(p, q) consists of the points corresponding to two consequent gradient

flows: first from p to some other critical point s, µ(p) > µ(s) > µ(q), and then from s to q:

∂M(p, q) = ∪s M(p, s) ×M(s, q).

Similarly, codimension 2 boundary consists of the points corresponding to three consequent

flows p → s → r → q with µ(p) > µ(s) > µ(r) > µ(q), etc. Including all these boundary

components we obtain a compact oriented manifold M (p, q).

Now we define Witten complex C∗(M, h) as a free abelian group generated by the set

of critical points of h:

Ck(M, h) = ⊕µ(p)=kZZ · [p].

Furthermore, we define ∂:Ck(M, h)→ Ck−1(M, h) via the sum over gradient trajectories

(counted with signs):

∂[p] =∑

µ(q)=k−1

#M(p, q) [q].

Then, the Morse-Thom-Smale-Witten theorem says that ∂2 = 0 and:

H∗(C∗(M, h)) = H∗(M, ZZ). (5.3)

Therefore, the classification of critical points and gradient trajectories which represent

supersymmetric vacua and BPS solutions, respectively, can be addressed in terms of the

topology of M. In particular, one finds the following lower bound on the total number of

supersymmetric vacua:

∑rank Ci(M, h) ≥

∑rank Hi(M,R). (5.4)

This is the classical Morse inequality.

We conclude that by using Morse theory formulas (5.2) – (5.4) one can classify five-

dimensional systems discussed in sections 2 and 3. In what follows we illustrate these

methods in a number of interesting examples and, in particular, we count the number

of supersymmetric vacua computing the Betti numbers bi = rank Hi(M,R) of the cor-

responding scalar field manifolds. Even though the general formulas (5.2) – (5.4) were

derived for a compact manifold M, the Morse inequality (5.4) is expected to hold in a

23

wider class of examples, including certain non-compact manifolds MV and MH relevant

for supergravity. For example, let us consider N ≥ 2 five-dimensional gauged supergravity

interacting with a certain number of vector multiplets. As we will see in a moment, such

theories possess at most one supersymmetric critical point on every branch of the scalar

field manifoldMV , as long as interaction with other matter fields (e.g. hyper multiplets)

can be consistently ignored, and as long as the superpotential W is generic enough to be

considered as a good global Morse function.

First, consider examples with exactly N = 2 supersymmetry where scalar fields φA

take values in a real homogeneous cubic hypersurfaceMV = F = 1 defined by (3.1) in a

vector space parameterized by the fields XI with Minkowski signature [25]. We expect that

real cubics of this form have trivial topology. Even though we do not have a mathematical

proof of this fact, we argue as follows. Suppose, on the contrary, that MV is topologically

non-trivial. Then, there should exist at least two critical points, one of which must have

non-zero Morse index, 0 < µ < nV . However, as we explained in section 2, the existence

of such points would violate the c-theorem [11] and contradict our original assumtion that

MV is a Riemannian manifold with a positive-definite metric.

Now let’s see that scalar field manifoldMV in N > 2 five-dimensional gauged super-

gravity is also topologically trivial. This result immediately follows from the fact14 that in

supergravity theories with more supersymmetry the spaceMV can always be represented

as a quotient space of a non-compact group G by its maximal compact subgroup H [42].

Note, that G can have time-like directions which would be inconsistent if we did not divide

by H that makes the metric on the quotient spaceMV = G/H positive-definite. Another

effect of the quotient by H is that the space G/H is topologically trivial unless we divide

further by a discrete symmetry group, e.g. U-duality group15. This is one way to make

the topology of M topologically non-trivial.

The second possibility to get theories with multiple supersymmetric vacua is to take

a space of scalar fields M with more than one branch. For instance, an important family

of theories is based on the target spaces of the following simple form:

MV = SO(n, 1)/SO(n) : Since SO(n) is the maximal compact subgroup in the non-

compact group SO(n, 1) the quotient space MV is equivalent, up to homotopy, to a set

14 We thank R. Kallosh and N. Warner for explaining this to us.15 We are grateful to E. Witten for pointing out that a quotient by a discrete group may lead

to interesting non-trivial topology of MV .

24

two points. The number of points comes from the number of disconnected components

in the non-compact group SO(n, 1) which looks like a hyperbolic space. Therefore, ir-

regardless of the value of n in this class of models we find the following Betti numbers

bi = rank Hi(MV ,R):

b0 = 2,

bi = 0, i > 0.

From the Morse inequality (5.4) it follows that the corresponding supergravity theories

with generic W possess at least two UV attractive supersymmetric vacua. By numerical

computations, one can verify that there are exactly two vacua of UV type. We remark that

there are no smooth domain walls because the two vacua belong to different disconnected

branches.

MV = SO(n − 1, 1) × SO(1, 1)/SO(n− 1) : Once again, in this example we divide

a non-compact group by its maximal compact subgroup, so that the resulting space is

isomorphic to a set of points for the reason explained above. This time we get 4 points,

one for every disconnected component of SO(n − 1, 1)× SO(1, 1). For the Betti numbers

of MV we obtain:

b0 = 4,

bi = 0, i > 0.

Similar to the previous example, there are at least four UV attractive vacua which can not

be smoothly connected by BPS domain walls.

These are examples of symmetric spaces which typically appear as scalar field mani-

folds in N = 2 five-dimensional supergravity theories and also in models with additional

(super-)symmetry structure [49], as we mentioned earlier.

It is important to stress here that it is crucial for the height function h to be globally

defined over the entire target space M. For example, this assumption breaks down in a

very important class of models corresponding to M theory compactification on Calabi-Yau

three-folds. In these models MV is just the Kahler structure moduli space of the Calabi-

Yau manifold. Since the Kahler structure moduli spaces usually have trivial topology, one

might naively conclude from (5.3) that there is only one vacuum (a UV attractive fixed

point) and no non-trivial domain walls. However, in general,MV consists of several Kahler

cones separated by the walls where certain algebraic curves in the Calabi-Yau space shrink

to zero size. Local anomaly arguments in heterotic M theory suggest that G-flux should

25

jump while crossing a Kahler wall [50]. In fact, passing through a flop transition point the

second Chern class changes and, therefore, the 5-brane charge induced in the boundary

field theory also jumps. Since the total 5-brane charge should be conserved (and equal to

zero in the compact Horava-Witten setup) some αI also have to jump, as if the flop curves

effectively carry a magnetic charge [50]. Although the superpotential (3.9) passes these

curves smoothly, because the corresponding XI vanishes there, its (second) derivatives

jump once we cross a Kahler wall, and the corresponding h is not a globally defined height

function.

A similar result occurs in gauged supergravity theories coupled to hyper multiplets.

In this case supersymmetry implies that scalar fields from hyper multiplets parameterize

a quaternionic manifoldMH of negative curvature [51]:

R = −8(n2H + 2nH)

Even though this condition is not as much restrictive as the supersymmetry condition

(3.8) in the case of vector multiplets, known examples of quaternionic homogeneous coset

spaces that may serve as hypermultiplet target manifolds are typically non-compact and

topologically trivial:

MH = SO(n, 4)/SO(n) × SO(4) : Similar to the exampleM = SO(n, 1)/SO(n), this

quaternionic space is contractible since16 SO(n)×SO(4) is the maximal compact subgroup

of SO(n, 4). Therefore, MH is homologous to a set of two points:

b0 = 2,

bi = 0, i > 0.

The same result we find forMH = SU(n, 2)/SU(n)×SU(2). As we mentioned earlier,

a simple way to obtain models where scalar fields take values in a topologically non-trivial

manifold MH is to devide by a discrete group which, for example, may be a subgroup of

the isometries of MH . For example, if we have only the universal hypermultiplet, n = 1,

the coset space MH = SU(1, 2)/U(1) × SU(2) is a quaternionic Kahler manifold, where

the Kahler potential can be written as K(S,C) = − log(S + S − 2(C + C)2). It has two

abelian isometries corresponding to shifts of the complex scalar fields S and C, S → S+ ia

16 Stricty speaking, S(O(n)×O(4)) is the maximal compact subgroup of SO(n, 4).

26

and C → C + ib. To get a topologically non-trivial manifold, we can consider a quotient

space:

MH/ZZ2 =SU(1, 2)

U(1) × SU(2)× ZZ2

where the action of ZZ2 is equivalent to identification S ∼ S+ ia and C ∼ C+ ib for integer

numbers a and b. Supergravity theory coupled to a hypermultiplet based on the resulting

quotient space is expected to have at least∑

i bi(MH/ZZ2) = 4 supersymmetric vacua.

Before we conclude this section, let us remark that due to its construction, the super-

potential or height function may not depend on all hyper-scalars and the chosen gauged

isometry of MH determines the scalars on which the superpotential depends. Hence,

for a given superpotential, obtained by a specific gauging, a mixed critical point with

0 < µ < nH may appear as a “good” UV or IR fixed point. Only the critical points

with µ = 0 and µ = nH are “gauge-independent” and appear in all gaugings as UV and

IR fixed points. Assuming that Morse inequalities are saturated, every component of Mhas exactly one UV critical point with µ = 0 and at most one IR point with µ = nH if

M is a compact manifold17. The values of all scalar fields are fixed in these two critical

points. Additional (mixed) critical points can be stable under the UV/IR scaling only if

the superpotential has (bad) flat directions.

6. Examples: SL(3) Symmetric Coset Spaces

Let us consider in more detail a specific example where the quotient space which

appears in N = 2 five-dimensional supergravity is associated with Jordan algebras of the

form:

M =Str0(J)

Aut(J)

where Str0(J) is the reduced structure group and Aut(J) is the automorphism group of

a real unital Jordan algebra of degree 3. A simple example based on irreducible J is

M = SL(3,R)/SO(3), which is a three-dimensional analog of the Lobachewsky plane,

SL(2,R)/SO(2). Like its two-dimensional analog, the space M is contractible because

a semisimple Lie group SL(n,R) is isomorphic to its maximal compact subgroup SO(n).

After we divide by the latter we get a point, up to homotopy:

b0 = 1 (6.1)

17 It should be stressed, however, that the authors do not know whether supergravity theories

based on compact scalar field manifolds M exist or not.

27

bi = 0, i > 0.

So, we come to the conclusion that the STU model with generic superpotential has a single

supersymmetric vacuum. In the rest of this section our goal will be to identify the physics

of this vacuum.

There are two generalizations of this example over complex numbers and quaternions.

In any case we divide SL(3, F ), where F = R,C,H is the base field in question, by

its maximal compact subgroup. Although the resulting quotient space has trivial topol-

ogy (6.1) for all ground fields F , the physics is different. Namely, we claim that three

supersymmetric vacua corresponding to various choices of F describe AdS5 dual of N = 4

super-Yang-Mills perturbed by mass terms which preserve different subgroups of SO(6)

R-symmetry.

In order to see that a scalar vev. gives rise to a mass deformation, one has to find

scaling dimension ∆(i) of the corresponding operator O(i) in the boundary theory. Using

the standard formula (2.13) we get ∆(i) = 2 which allows us to identify O(i) with a mass

term. In general, a mass term is specified by a bosonic symmetric 6 × 6 matrix and a

fermionic symmetric 4 × 4 matrix. In our three exampes, however, these matrices have

additional symmetries reminiscent of N = 4 super-Yang-Mills theory with R-symmetry

twists [52]. In particular, at special values of twists, where the corresponding phases are

equal to (−1), extra hypermultiplets become massless18, and we expect to get a four-

dimensional superconformal theory 19.

Below we discuss in more details the case where F = C. In particular, we solve the flow

equations (2.7) for the coset space SL(3,C)/SU(3). This coset manifold is parameterized

by 8 non-trivial scalar fields with the intersection form [15]

F = STU − S|X|2 − T |Y |2 − U |Z|2 + 2 Re(XY Z) (6.2)

18 To see this, it is convenient to think of gauge theories with R-symmetry twists as compact-

ifications of five-dimensional gauge theories on a circle with twisted boundary conditions on S1

[52]. The masses of Kaluza-Klein states are given by m = n+ 1/2 + qW where W is the value of

the Wilson lines of the gauge field and q is the charge of a given mode. Note that for any n ∈ ZZ

we get a whole hyper-multiplet. Moreover, for n = 0 and −1 and W = 1/2 and q = +1 and −1

we find two massless hypermultiplets.19 We thank Ori Ganor for discussions on this point.

28

where X,Y and Z are complex and S, T and U are real. The unique solution of the

attractor equations [36]: ∂IF = e−2U HI , which solves the flow equations (2.7) (see [30])

is given

S = (HTHU −1

4|HX |2) e−4U , X =

1

4(HYHZ − 2HXHS) e−4U ,

T = (HSHU −1

4|HY |2) e−4U , Y =

1

4(HXHZ − 2HYHT ) e−4U ,

U = (HSHT −1

4|HZ |2) e−4U , Z =

1

4(HXHY − 2HZHU ) e−4U

(6.3)

where HI is a set of harmonic functions

HI = hI + 6αI y

which are real for the S, T,U components and complex for the X,Y,Z components. For

the metric we take the ansatz:

ds2 = e2U[− dt2 + d~x2

]+ e−4Udy2 (6.4)

where the function e−2U is obtained from the requirement F = 1:

e6U = HSHTHU −1

4

(HS |HX |2 +HT |HY |2 +HU |HZ |2

)+

1

4Re(HXHYHZ) . (6.5)

The scalar fields are defined by F = 1 and we may consider, for example, the ratios:

φA = TS , US , XS , YS , ZS . The uniqueness of the solution fits very well with our expectation

from Morse theory, that this coset allows only a unique critical point. At the critical point

the space time becomes AdS5 as y → +∞ with the negative cosmological constant:

Λ = −(e4U/y2)y→+∞

In the case of a unique critical point with a negative cosmological constant, it is natural

to ask what is the four-dimensional field theory dual to this AdS5 vacuum.

Notice that the scalar fields φA defined as ratios of harmonic functions stay finite in

the AdS vacuum and approach their critical values. Having the explicit solution, we can

also calculate the supergravity effective action. As given by (2.6), the BPS nature ensures

vanishing of the bulk term and the surface part yields20:

Seff =2

3

(∂ye

6U)y=+∞

=(− 2Λy2 + a1 y

)y→∞

+ a2. (6.6)

20 Note the different coordinate system.

29

As expected, this action has singular terms and a finite part. The leading singularity scales

with the cosmological constant (AdS volume) while the subleading term a1 and the finite

term a2 can be obtained by inserting the harmonic functions (6.5). The divergent part will

be subtracted by the renormalization in the field theory and the finite part will give the

renormalized effective action; in our approximation we see only potential or mass terms.

From the RG-flow point of view this AdS vacuum corresponds to an UV fixed point.

While moving towards negative y the warp factor e2U decreases monotonically (according

to c-theorem) and we approach the IR region in the fields theory. Because e6U is negative

at y = −∞, we have to pass a zero at some finite value of y, which is the singular end-point

of the RG flow. Like in any other case with vector multiplets only, the absence of an IR

fixed point forces the solution to run into a singularity.

On the other hand, from the string theory perspective this solution corresponds to a

5-brane wrapping a holomorphic 2-cycle, namely a torus T 2. Once again, we point out

the analogy with the construction of gauge theories with R-symmetry twists from (2,0)

theory on a torus in the limit where the size of the torus and the values of twists tend to

zero (Vol(T 2) → 0 and α → 0) while their ratio α/Vol(T 2) remains finite and defines a

mass scale in the resulting theory [52]. From this construction it is clear that at the special

values of twists α, where the theory becomes superconformal, it must be dual to AdS5×S5

perturbed by a dimension-8 operator (proportional to Vol(T 2)) and dimension-2 operators

(proportional to α), similar to the AdS5 vacuum of SL(3) coset spaces we found.

It is reasonable to put another (5-brane) source at some place where the warp factor

is still positive, say y = 0. This extra source appears as a non-trivial right-hand side of

the harmonic equations:

∂2HI ∼ αIδ(y)

where αI is the component of the 5-brane charge related to a basis of (1, 1)-forms ωI .

There are two possibilities: in the first option we continue in a symmetric way through the

source, which implies the replacement HI → hI +6αI |y| and is equivalent to a sign change

in the flux vector αI while passing the source at y = 0. This case appears in the ZZ2 orbifold

of the Horava-Witten setup compactified to 5 dimensions [23]. But we may also consider

the case where the flux jumps from zero to a finite value, i.e. on the side behind the source

we can set αI = 0, which is equivalent to the replacement H → hI + 6αI12(y+ |y|), so that

HI is constant for negative y and the space time is flat.

By adding this source, we cut off the (singular) part of space-time and glue instead

an identical piece (Z2 symmetric) or flat space (vanishing flux on one side). If one wants

30

to discuss a RS-type scenario, one may also cut off the regular AdS part and keep on both

sides the naked singularity. In the first case the source is the standard positive tension

brane generating an AdS space on both sides, whereas in the second case a negative tension

brane is accompanied with a naked singularity. For a more detailed discussion of sources

see [53].

Acknowledgments

We are grateful Alexander Chervov, Ori Ganor, Brian Greene, Renata Kallosh, Eric

Sharpe, Gary Shiu, Andrew Strominger, Nicholas Warner, and Edward Witten for help-

ful discussions and comments. The work of K.B. was partly done at the Theory group

of Caltech and is supported by a Heisenberg grant of the DFG and by the European

Commission RTN programme HPRN-CT-2000-00131. S.G. is supported in part by the

Caltech Discovery Fund, grant RFBR No 98-01-00327 and Russian President’s grant No

00-15-99296.

31

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